Question

The cost of televisions exhibits huge variation \(-\) from \(\$ 100-200\) for a standard TV to \(\$ 8,000-10,000\) for a large plasma screen TV. Consumer Reports gives the prices for the top 10 LCD high definition TVs (HDTVs) in the 30 - to 40 -inch category: $$ \begin{array}{lc} \text { Brand } & \text { Price } \\ \hline \text { JVC LT-40FH96 } & \$ 2900 \\ \text { Sony Bravia KDL-V32XBR1 } & 1800 \\ \text { Sony Bravia KDL-V40XBR1 } & 2600 \\ \text { Toshiba 37HLX95 } & 3000 \\ \text { Sharp Aquos LC-32DA5U } & 1300 \\ \text { Sony Bravia KLV-S32A10 } & 1500 \\ \text { Panasonic Viera TC-32LX50 } & 1350 \\ \text { JVC LT-37X776 } & 2000 \\ \text { LG 37LP1D } & 2200 \\ \text { Samsung LN-R328W } & 1200 \end{array} $$ a. What is the average price of these 10 HDTVs? b. What is the median price of these 10 HDTVs? c. As a consumer, would you be interested in the average cost of an HDTV? What other variables would be important to you?

Short answer

Answer: The average price of these HDTVs is $1985, and the median price is $1900. The average cost serves as a general indicator of the price range, but it is not the only factor a consumer should consider. Other important factors include screen size, picture quality, brand reputation, additional features, energy consumption, and after-sales service.

Step-by-step solution

Calculate the total cost of these HDTVs

First, we will find the total cost of these HDTVs by adding the prices of each of the 10 HDTVs: $$ 2900 + 1800 + 2600 + 3000 + 1300 + 1500 + 1350 + 2000 + 2200 + 1200 $$

Calculate the average price of these HDTVs

To find the average price, we will divide the total cost found in step 1 by the total number of HDTVs, which is 10: $$ \text{Average Price} = \frac{\text{Total Cost}}{\text{Number of HDTVs}} = \frac{2900 + 1800 + 2600 + 3000 + 1300 + 1500 + 1350 + 2000 + 2200 + 1200}{10} $$ Find the value to get the average price.

Arrange the prices in ascending order

To find the median price, we first need to arrange the prices in ascending order: $$ 1200, 1300, 1350, 1500, 1800, 2000, 2200, 2600, 2900, 3000 $$

Find the median price of these HDTVs

Since the total number of HDTVs is even (10), we take the middle two values (5th and 6th prices) and find their average: $$ \text{Median Price} = \frac{\text{5th Price} + \text{6th Price}}{2} = \frac{1800 + 2000}{2} $$ Find the value to get the median price.

Discuss usefulness of average cost

As a consumer, the average cost of an HDTV can provide some information about the general price range of HDTVs in the market. However, other factors must also be considered when deciding to purchase an HDTV, such as screen size, picture quality, brand reputation, additional features, energy consumption, and after-sales service. All of these factors may greatly affect the overall satisfaction of the consumer.

Key concepts

Average Price Calculation

Calculating the average price of a product is an essential skill in statistical data analysis, often used to estimate the central value within a data set. In our exercise, to find the average price of the top 10 LCD HDTVs, we must first sum up all individual TV prices. With a total accumulated, the average price is simply this total divided by the number of TVs, hence providing a single value that represents the 'typical' cost one might expect to pay for an HDTV in this category.

Mathematically, the average (also known as the arithmetic mean) is defined as:
\[\text{Average Price} = \frac{\text{Total Cost of HDTVs}}{\text{Number of HDTVs}}\]
Using the numbers from our exercise, the average price calculation yields a value that reflects the cost spread of the HDTVs, allowing a consumer to understand what price is common or typical within a given market or set of products.

Median Price Calculation

Another important measure in data analysis is the median price, which represents the middle value of an ordered data set. It's particularly useful in providing a robust measure of central tendency, especially when the data includes outliers that could skew the average. To find the median, the prices must be listed in ascending or descending order, and since we have an even number of data points (10 HDTVs), the median is the average of the two middle numbers—fifth and sixth prices when arranged from lowest to highest.

Here's the formula used:
\[\text{Median Price} = \frac{\text{5th Price} + \text{6th Price}}{2}\]
The median offers a very different perspective compared to the average. It's often a preferred measure when shopping, as it can point to what is more commonly available pricing, regardless of extremely high-priced or low-priced items.

Data Interpretation

While average and median calculations provide numerical insights, the art of data interpretation involves looking beyond the numbers to understand the context and implications. What does the average price tell a consumer? Is it skewed by uniquely expensive models? Does the median offer a better representation of what one might pay? Consumers might find the median price more applicable, as it tends to represent the 'middle ground' which is less affected by the extreme prices in the variability of HDTVs.

In addition to the central tendency measures, consumers should consider other essential variables affecting the decision-making process, such as brand reputation, screen size, resolution, smart features, warranty, and customer reviews. The ability to interpret this data effectively allows a consumer to make a well-informed choice that aligns with their preferences and budget, ensuring satisfaction with their purchase.